35 research outputs found
Quantifying structure in networks
We investigate exponential families of random graph distributions as a
framework for systematic quantification of structure in networks. In this paper
we restrict ourselves to undirected unlabeled graphs. For these graphs, the
counts of subgraphs with no more than k links are a sufficient statistics for
the exponential families of graphs with interactions between at most k links.
In this framework we investigate the dependencies between several observables
commonly used to quantify structure in networks, such as the degree
distribution, cluster and assortativity coefficients.Comment: 17 pages, 3 figure
Nonparametric Information Geometry
The differential-geometric structure of the set of positive densities on a
given measure space has raised the interest of many mathematicians after the
discovery by C.R. Rao of the geometric meaning of the Fisher information. Most
of the research is focused on parametric statistical models. In series of
papers by author and coworkers a particular version of the nonparametric case
has been discussed. It consists of a minimalistic structure modeled according
the theory of exponential families: given a reference density other densities
are represented by the centered log likelihood which is an element of an Orlicz
space. This mappings give a system of charts of a Banach manifold. It has been
observed that, while the construction is natural, the practical applicability
is limited by the technical difficulty to deal with such a class of Banach
spaces. It has been suggested recently to replace the exponential function with
other functions with similar behavior but polynomial growth at infinity in
order to obtain more tractable Banach spaces, e.g. Hilbert spaces. We give
first a review of our theory with special emphasis on the specific issues of
the infinite dimensional setting. In a second part we discuss two specific
topics, differential equations and the metric connection. The position of this
line of research with respect to other approaches is briefly discussed.Comment: Submitted for publication in the Proceedings od GSI2013 Aug 28-30
2013 Pari
Dynamical and Stationary Properties of On-line Learning from Finite Training Sets
The dynamical and stationary properties of on-line learning from finite
training sets are analysed using the cavity method. For large input dimensions,
we derive equations for the macroscopic parameters, namely, the student-teacher
correlation, the student-student autocorrelation and the learning force
uctuation. This enables us to provide analytical solutions to Adaline learning
as a benchmark. Theoretical predictions of training errors in transient and
stationary states are obtained by a Monte Carlo sampling procedure.
Generalization and training errors are found to agree with simulations. The
physical origin of the critical learning rate is presented. Comparison with
batch learning is discussed throughout the paper.Comment: 30 pages, 4 figure
Optimal measures and Markov transition kernels
We study optimal solutions to an abstract optimization problem for measures,
which is a generalization of classical variational problems in information
theory and statistical physics. In the classical problems, information and
relative entropy are defined using the Kullback-Leibler divergence, and for
this reason optimal measures belong to a one-parameter exponential family.
Measures within such a family have the property of mutual absolute continuity.
Here we show that this property characterizes other families of optimal
positive measures if a functional representing information has a strictly
convex dual. Mutual absolute continuity of optimal probability measures allows
us to strictly separate deterministic and non-deterministic Markov transition
kernels, which play an important role in theories of decisions, estimation,
control, communication and computation. We show that deterministic transitions
are strictly sub-optimal, unless information resource with a strictly convex
dual is unconstrained. For illustration, we construct an example where, unlike
non-deterministic, any deterministic kernel either has negatively infinite
expected utility (unbounded expected error) or communicates infinite
information.Comment: Replaced with a final and accepted draft; Journal of Global
Optimization, Springer, Jan 1, 201
Universal neural field computation
Turing machines and G\"odel numbers are important pillars of the theory of
computation. Thus, any computational architecture needs to show how it could
relate to Turing machines and how stable implementations of Turing computation
are possible. In this chapter, we implement universal Turing computation in a
neural field environment. To this end, we employ the canonical symbologram
representation of a Turing machine obtained from a G\"odel encoding of its
symbolic repertoire and generalized shifts. The resulting nonlinear dynamical
automaton (NDA) is a piecewise affine-linear map acting on the unit square that
is partitioned into rectangular domains. Instead of looking at point dynamics
in phase space, we then consider functional dynamics of probability
distributions functions (p.d.f.s) over phase space. This is generally described
by a Frobenius-Perron integral transformation that can be regarded as a neural
field equation over the unit square as feature space of a dynamic field theory
(DFT). Solving the Frobenius-Perron equation yields that uniform p.d.f.s with
rectangular support are mapped onto uniform p.d.f.s with rectangular support,
again. We call the resulting representation \emph{dynamic field automaton}.Comment: 21 pages; 6 figures. arXiv admin note: text overlap with
arXiv:1204.546
U.S. stock market interaction network as learned by the Boltzmann Machine
We study historical dynamics of joint equilibrium distribution of stock
returns in the U.S. stock market using the Boltzmann distribution model being
parametrized by external fields and pairwise couplings. Within Boltzmann
learning framework for statistical inference, we analyze historical behavior of
the parameters inferred using exact and approximate learning algorithms. Since
the model and inference methods require use of binary variables, effect of this
mapping of continuous returns to the discrete domain is studied. The presented
analysis shows that binarization preserves market correlation structure.
Properties of distributions of external fields and couplings as well as
industry sector clustering structure are studied for different historical dates
and moving window sizes. We found that a heavy positive tail in the
distribution of couplings is responsible for the sparse market clustering
structure. We also show that discrepancies between the model parameters might
be used as a precursor of financial instabilities.Comment: 15 pages, 17 figures, 1 tabl